Second-Degree Price Discrimination for Information Goods Under Nonlinear Utility Functions

Transcription

1 Second-Degree Price Discrimination for Information Goods Under Nonlinear Utility Functions Hemant K. Bhargava The Smeal College of Business Pennsylvania State University 342 Beam Building University Park, PA Tel: (814) Fax: (814) Vidyanand Choudhary Graduate School of Industrial Administration Carnegie Mellon University 5000 Forbes Avenue Pittsburgh, PA Tel: (412) Fax: (412) Abstract This paper studies second degree price discrimination for information goods. Prior research in the context of traditional goods (where marginal costs are convex as a function of product quality) shows that a firm can increase profits by offering vertically differentiated products to heterogeneous consumers. Some researchers have obtained the converse result for information products, assuming negligible marginal costs which vary little with product quality. Earlier models have used specific forms of the consumer utility function (most commonly a utility function that is linear in type and product quality). We model this problem using a generalized utility function where the utility is monotonically increasing with product quality and consumer type. We find that for information goods, price discrimination is profitable only in markets where high-value consumers benefit relatively more than low-value consumers from increases in quality. 1 Introduction Research in pricing and economic theory ([11]) has shown the effectiveness of price discrimination and product differentiation strategies when a firm markets its products to heterogeneous consumers. When consumers differ in their willingness to pay for a product, firms can extract greater profits by segmenting consumers. This can be achieved either by offering multiple product qualities in the market and allowing consumers to self-select (second-degree price discrimination [5, 6, 13]) or by isolating consumers and offering the product at different prices to each consumer (first-degree price discrimination) or offering different prices for different groups (third-degree price discrimination). This paper is concerned with second-degree price discrimination, which is a useful pricing strategy when firms cannot isolate consumers individually or into groups. We continue prior inquiries into whether second-degree price discrimination is an effective strategy even in the case of (digitizable) information goods. A distinguishing characteristic of such goods is the unusual property that the marginal cost of production is near zero or, more generally, invariant with product quality [8]. As with other goods, intuition suggests that seconddegree price discrimination would allow a monopolistic seller of information goods to increase profits by increasing the fraction of buyers. However, recent analytical models have demonstrated that second-degree price discrimination is not profitable in the case of information goods. Meyer [4], Mendelson and Jones [3], and Varian [12]) have shown - under fairly restrictive modeling assumptions - that second-degree price dis /01 $10.00 (c) 2001 IEEE 1

2 crimination does not work in the case of information goods. Salant [7] examined the connections between the works of Stokey on intertemporal price discrimination [10], Spence on quantity-dependent price discrimination [9] and Mussa and Rosen on quality-based price discrimination [6]. Using a linear utility function and a finite set of consumer types, he showed that price discrimination would be suboptimal when the marginal cost function is linear (in the Mussa and Rosen problem; analogously, a linear utility function in Spence s or Stokey s problems). Bhargava and Choudhary [1] have obtained the same result for a generalized consumer distribution. Thus, prior theoretical work suggests that a monopolistic seller of information goods would not be able to use price discrimination profitably; any profit-maximizing prices would cause all consumers to buy a single product. Bhargava and Choudhary [1] attribute this result to the unique marginal cost structure for information goods i.e., that marginal costs of information goods do not increase significantly with product quality [8]. The above findings assume a linear utility function where the utility obtained by a consumer is a linear function of the consumer type θ and product quality q. This follows modeling tradition in price discrimination and product differentiation, but raises the conjecture that this counter-intuitive result is dependent on the specific utility function (U(θ, q) =θ q)employed in these models. In related work, Bhargava and Choudhary [2] examine pricing strategies for information intermediaries and find that price discrimination is indeed an optimal strategy. In this, model the consumer s valuation of the intermediary s service is of the form θ q + k( ) wherethetermk( )representsan additional benefit that buyers obtain due to the aggregation (of sellers) effect provided by intermediaries. In their model, k( ) is a function of parameters that can be controlled by the intermediary. It is easily seen, however, that even when k is a constant, the fundamental result remains the same: price discrimination is optimal. What do we have so far? We know that under standard economic modeling assumptions (U(θ, q) = θ q), price discrimination is not optimal for monopolistic sellers of information goods, but that this result is inverted by a small perturbation in the utility function. This raises the questions: how sensitive is this important result to the nature of the utility function? and what does this mean for purveyors of information goods, of which there is a rapidly growing number in the information economy? In this paper, we analyze the effectiveness of second-degree price discrimination under a generalized utility function. We determine classes of utility functions for which price discrimination would or would not be profit-maximizing. Specifically, we find that when the consumer s valuation function U(θ, q) is separable into the form g(θ) h(q) for arbitrary functions g and h, price discrimination is not profit-maximizing. We present our mathematical model in 2 and develop results about price discrimination in 3. We interpret and discuss our results in a concluding section. 2 Model Formulation We begin by analyzing a two-product monopolist selling information goods to heterogeneous consumers, distinguished by their taste for service through a taste parameter θ [0, 1]. Each consumer s valuation of the product is determined by her taste parameter and the quality of the product. Thus, a consumer of type θ has willingness to pay U(θ, q) for one unit of a product of quality q, and zero for additional units. We assume that consumers are uniformly distributed along the taste parameter (θ) in the interval [0, 1]. The firm is cognizant of this aggregate distribution but unaware of each consumer s type. The firm offers the product at two levels of quality q L and q H to all consumers at prices p L and p H respectively. The quality levels q L and q H are assumed to be given exogenously. Consumers select the product that offers them the higher surplus. The utility function U(θ, q) is assumed to be monotonically increasing in θ and in q. WewriteU (θ, q), U (θ, q) for the first and second partial derivatives, respectively, of U(θ, q) with respect to θ and Û(θ, q), for the first partial derivative of U(θ, q) with respect to q. We assume that U(θ, q), U (θ, q) andû(θ, q) are positive, continuous, and dif /01 $10.00 (c) 2001 IEEE 2

3 ferentiable everywhere, with respect to θ. Further we assume that U(θ,q) U (θ,q) is increasing in θ. It can be verified that all concave functions and most convex functions satisfy this condition. Given prices p L and p H, there will exist indifferent consumer types θ L and θ i where a consumer of type θ i obtains equal surplus for both product qualities q H and q L (hence indifferent between the two qualities), and a consumer of type θ L obtains a zero surplus for product q L (hence indifferent between buying q L and not buying at all). Consumers with taste parameter in [θ i, 1] will choose to buy quality q H and consumers in [q L,q H ) will choose to buy q L. The firm s profit function is π = p H (1 θ i )+p L (θ i θ L ) (1) From the definition of indifferent types θ i and θ L we get p L = U(θ L,q L ) (2) p H p L = U(θ i,q H ) U(θ i,q L ) p H =U(θ i,q H ) U(θ i,q L ) +U(θ L,q L ) (3) We rewrite π using Eq. 2 and Eq. 3 and we have the firm s optimization problem: Max π θ i,θ L = (1 θ i )(U(θ i,q H ) U(θ i,q L ) +U(θ L,q L )) + (θ i θ L )U(θ L,q L ) subject to 0 θ L θ i 1 (4) 3 Analysis We analyze this model to determine the efficacy of price discrimination in maximizing the firm s profits. To determine optimal indifference points, we compute first and second partials of π with respect to θ i and θ L. Setting these partials to zero, we obtain conditions for the optimal indifference points θ L and θ i. Price discrimination will be profit-maximizing if and only if θ i >θ L. The profit function (Eq. 4) yields the first partial derivatives π θi = 1[U(θ i,q H ) U(θ i,q L )+U(θ L,q L )] +(1 θ i )[U (θ i,q H ) U (θ i,q L )] +U(θ L,q L ) (5) π θ L = (1 θ i )U (θ L,q L ) U(θ L,q L ) +(θ i θ L )U (θ L,q L ) (6) Setting the partials in Eq. 5 and Eq. 6 equal to zero and simplifying, we get (1 θ i ) = U(θ i,q H ) U(θ i,q L ) U (θ i,q H ) U (7) (θ i,q L ) (1 θ L ) = U(θ L,q L ) U (8) (θ L,q L ) We represent the right side of Eq. 7 and 8 as A(θ) and B(θ) respectively: A(θ) = U(θ i,q H ) U(θ i,q L ) U (θ i,q H ) U (θ i,q L ) B(θ) = U(θ L,q L ) U (θ L,q L ) We introduce the following notation to indicate the solutions of Eq. 7 and 8: θi = Solve[A(θ)+θ=1] (9) θl = Solve[B(θ)+θ= 1] (10) It can be verified that when B(θ) is increasing in θ, the cardinality of {θl } is at most 1. Specifically, there is a unique feasible value of θl when B(θ) 1at θ= 0. There is no feasible solution in the trivial case when B(θ) > 1atθ= 0 which violates the boundary θ L 0; in this case we set θl =0. As we assumed earlier, U(θ,q) U (θ,q) is an increasing function of θ and B(θ) = U(θL,qL) U (θ L,q L), hence B(θ) is increasing in θ. This establishes that θl is unique. Further it can be verified that under these assumptions, θl satisfies second-order conditions for maxima. 3.1 Conditions for Optimality of Price Discrimination Since θi and θl represent the indifference points, we see that the firm can profitably employ price discrimination if and only if θi >θ L. We now derive conditions /01 $10.00 (c) 2001 IEEE 3

4 under which this happens (or not), and see that these conditions are related to how the ratio of valuations between high and low quality products changes with consumer type. 1) When the denominator of A(θ) is negative, i.e. (U (θ, q H ) U (θ, q L )) < 0, we find that there is no solution to the equation A(θ)+θ = 1. To see this, note that the numerator of A(θ) is positive since Û(θ, q) is assumed to be positive. This implies that A(θ) < 0, hence there is no solution in the feasible region of θ [0, 1]. In this case the firm sells only one product and there is no price discrimination. 2) When the denominator of A(θ) is positive, i.e. (U (θ, q H ) U (θ, q L )) > 0: When A(θ) <B(θ) there may be multiple θi solutions but they will all be greater than θl. This would guarantee that price discrimination is optimal. Examining Eq. 9 and Eq. 10, weseethat (θi >θl) U(θ, q H) U (θ, q H ) < U(θ, q L) U (θ, q L ) Manipulating and rewriting this equation, we see that price discrimination is an optimal strategy when the utility function satisfies either of the following conditions that are all equivalent to A(θ) <B(θ): U(θ, q H ) U(θ, q L ) U(θ, q H ) U(θ, q L ) U(θ, q H ) θ U(θ, q L ) < U (θ, q H ) U (θ, q L ) < U(θ + θ, q H) U(θ + θ, q L ) θ 0 > 0 (11) We interpret Eq. 11 as the ratio of valuations at q H and q L, and therefore can state that When the ratio of valuations increases in θ then price discrimination is an optimal strategy. 3) The converse of the point we make above is also true. When A(θ) >B(θ) there may be multiple θi solutions but they will all be less than θl. This implies that the constraint θ i θ L is tight, i.e. θ i = θ L. This implies that the firm does not sell q L to anyone (the fraction of consumers buying q L is given by θ i θ L, which is equal to zero). This would guarantee that price discrimination is not optimal. As stated above in item one, when the denominator of A(θ) is negative, i.e. U (θ, q H ) U (θ, q L ) < 0, we find that price discrimination is not optimal, it also happens to be the case that the ratio of valuations is decreasing with respect to θ whenever the denominator of A(θ) is negative. Therefore, we can summarize our findings by saying that whenever the ratio of valuations is increasing in θ, it is optimal for the firm to price discriminate when the ratio of valuations is decreasing in θ, it is not optimal to price discriminate. 4 Conclusion We have derived conditions under which price discrimination would be a profitable strategy for a monopolist selling information goods to heterogeneous consumers. Our analysis assumes that consumers are uniformly distributed along the set of consumer types. Based on results from prior research, we expect that the results derived in this paper are not highly sensitive to this assumption. A formal analysis of this assumption is beyond the scope of this paper. Our other key assumption is that the term U(θ,q) U (θ,q) is increasing in θ. This assumption is satisfied by all concave functions. It is also satisfied by many convex functions, except ones that have a sharp increase. This assumption is sufficient, although not necessary, to guarantee uniqueness of the indifference point θ L. We find that) in markets where the ratio of valuations is higher for high-value consumers ( U(θ,qH) U(θ,q L) relatively to the lower-value consumers, second-degree price discrimination will cause market segmentation and improve profit. Under such circumstances, a firm should offer multiple product qualities. In the alternative case, when the ratio of valuations is smaller for high-value consumers relatively to the lower-value consumers, price discrimination is not an optimal strategy for sellers of information goods. In such situations, we turn to Bhargava and Choudhary [1] for an explanation Introduction of a low quality product into the market has two effects: it causes some low-value consumers (who would not otherwise have purchased a product) to enter /01 $10.00 (c) 2001 IEEE 4

5 the market, but it also causes some highvalue consumers to shift to the low quality product...when marginal costs are negligible, the reduction in revenue is equivalent to a reduction in profit... The overall reduction in profit makes price discrimination suboptimal. To further understand the circumstances under which price discrimination may be an optimal strategy, we offer two additional perspectives on our results. The condition ratio of valuations is increasing in θ isnot satisfied by any utility function U(θ, q) that is separable into the form g(θ) h(q) for arbitrary functions g and h. To see this, first note that the condition ratio of valuations is increasing in θ is the same as U (θ, q H ) U (θ, q L ) > U(θ, q H) U(θ, q L ) However, both terms in the statement above resolve to h(qh) h(q L) when the utility function is separable. Therefore, price discrimination is not optimal when we can write U(θ, q) =g(θ) h(q) For example, this is true for the commonly assumed utility function U(θ, q) =θ q. In this case and in many other examples of commonly used utility functions, it is possible to express U(θ, q) as g(θ)h(q). The condition ratio of valuations is increasing in θ is satisfied, however, by the following simple variation on the function θ q U(θ, q) =(θ q)+k This kind of utility function would apply where all consumers get a basic benefit from using the product that is unaffected by q and θ. In this case offering a second lower-quality product improves the firm s profits. However, note that the additional benefit k obtained by consumers must come at no cost to the seller. This happens, e.g., when there are network benefits. In related work, Bhargava and Choudhary [2] have argued in the case of information intermediaries that the presence of aggregation benefits may lead to such a valuation function. We are continuing to explore other circumstances that would be compatible with market segmentation for information goods, and to develop further insights that offer better explanation and application of the mathematical results developed in this paper. Acknowledgements We gratefully acknowledge Rajiv Dewan and anonymous reviewers whose timely and constructive comments were useful in improving this paper. References [1] Bhargava, H. K., and Choudhary, V. Price discrimination and product differentiation in information goods. Tech. rep., Carnegie Mellon University, GSIA Working Paper , [2] Bhargava, H. K., and Choudhary, V. Economics of an electronic intermediary with aggregation benefits. Tech. rep., Carnegie Mellon University, GSIA Working Paper , [3] Jones, R., and Mendelson, H. Product and price competition for information goods. Tech. rep., Stanford University, [4] Meyer, D. W. Essays on Quality and Product Differentiation. PhD thesis, The University of Michigan, [5] Moorthy, K. S. Market segmentation, selfselection, and product line design. Marketing Science 3, 4 (Fall 1984), [6] Mussa, M., and Rosen, S. Monopoly and product quality. Journal of Economic Theory 18, 2 (August 1978), /01 $10.00 (c) 2001 IEEE 5

6 [7] Salant,S.W. When is inducing self-selection suboptimal for a monopolist? Quarterly Journal of Economics 104, 2 (May 1989), [8] Shapiro, C., and Varian, H. Information Rules : A Strategic Guide to the Network Economy. Harvard Business School Press, Boston, MA, [9] Spence, A. M. Multi-product quantitydependent prices and profitability constraints. Review of Economic Studies 47, 5 (October 1980), [10] Stokey, N. L. Intertemporal price discrimination. Quarterly Journal of Economics 93, 3 (August 1979), [11] Tirole, J. The Theory of Industrial Organization. MIT Press, Cambridge, MA, [12] Varian, H. Microeconomic Analysis, 3 ed. Norton, [13] Varian, H. Differential pricing and efficiency, August /01 $10.00 (c) 2001 IEEE 6